8618
Acknowledgment. This work was assisted financially by grants from the National Institutes of Health and the Agency for International Development and a stipend t o K. B. B. from the Stiftung fur Stipendien auf dem Gebiete der Chemie, Switzerland. E. J. Carey,* Konrsd B. Beeker, Ravi K. Varma Department of Chemistry, Haraard University Cambridge, Massachusetts 02138 Received September 8,1972
Symmetry Selection Rules for Transition States
Figure 2. CPK model of I, R = p C e H & H , N H C O , with s-cis arrangement of the AL31I4)-en-15-oneunit, antiplanar H-C(I2)C(13k-H unit, maximum contact between enone chain (extended C,H,,) and p-phenylphenylcarbamoyl group. In this conformation the 0-CO-N-C part of the urethane function is planar and the C-O-CO-N part is nonplanar, corresponding to amide delocalization in preference to ester delocalization. The s-cis enone unit contacts four adjacent carbons of the NHCsH, unit. Numerals refer lo carbon Dosition.
1.0, CHCl,), was carried out from the alcohol I, R = H, by reaction with the readily available p-phenylphenyl isocyanate, mp 58-58S08 (1.2 equiv), in dry tetrahydrofuran (4 ml/g of isocyanate) and triethylamine (1.2 equiv) at 25" for 3 hr (yield, >90%). For reduction the p-phenylphenylurethane I in tetrahydrofuranether (1 :4,40ml/g of I) at ca. - 13Oo8was treated dropwise with a 0.23 M solution in tetrahydrofuran of the reagent IV1* ( c a 30 ml/g of I), reaction was continued for 4 hr at - 130" and 2 hr at - 115" (ethanol-liquid nitrogen bath), and the product was isolated by extraction after quenching of the reaction mixture with methanol and 1 N-hydrochloric acid. The total yield of 11,' R = p-CsH5CeH4NHC0, was quantitative, and the ratio of 15s to 15R isomers in several runs was 92/8. For the synthesis of prostaglandins it was found to be expedient to utilize the product I1 directly and t o remove the small amount of 15R by-product at the stage of prostaglandin F b or 6 where the separation of 15s and 15R diastereomers is extremely simple. The conversion of 11, R = p-CsHSC8H4NHC0,to the re-7.1" (c 1.1, quired diol I1 [R = H,'* oil, [ C Y ] ~ ~ D CHC18)]was accomplished in >90% yield by hydrolysis using 1 M aqueous lithium hydroxide at 120" for 72 hr, extraction of the basic reaction mixture at 0" with etherethyl acetate ( 1 : l ) t o remove neutral and basic components, and relactonization by the addition of ethyl chloroformate (2 equiv) t o the aqueous phase which had been neutralized with carbon dioxide.'O The highly selective reduction of I, R = p-CsHsCsH4NHCO, could also be carried out with thexyl di-secbutylborohydride (88/12) and tri-sec-butylborohydride" (89jll). (8) M. 1.van Geldcren, Reel. Trroo. Chim. Poys-Bo& 52,969 (1933). (9) Cooling bath: n-pentane-liquid nitrogen; see R. E. Rondeau, f. Chem. Eng. Data, 11,124(1966).
(IO) This procedure for laetonization was developed in collaboration with Dr. A. Venkateswarlu in these laboratories. (11) See H. C. Brown and S. Krishnamurthy, J. Amer. Chem. Soc., 94,7159 (1972).
Journal of the American Chemical Society
/ 94:24
Sir: The transition state for a chemical reaction is the lowest possible potential energy barrier (assuming one exists) between reactants and products. At the transition state there must exist directions along which the potential energy decreases as the atoms are displaced toward reactants or products. In other words, the curvature of the potential energy along such a direction must be negative. The unit vector leading from the transition state toward products along the direction of most negative curvature we define as the transition uec/or. It can be shown that this is the eigenvector of the force constant matrix F corresponding t o the lowest eigenvalue, that this eigenvalue is negative, and that F can have no other negative eigenvalues.' Moreover, the transition vector, like all eigenvectors of F, must belong to one of the irreducible representations of the point group of the transition state.% The purpose of this communication is to demonstrate the existence of selection rules governing the symmetry properties of the transition vector and, in fact, for a broad class of reactions, governing the structural symmetry of the transition state itself. These selection rules, presented in the form of three theorems, follow from group theoretical and geometric considerations alone. Consequently, they are valid for any potential surface which shows symmetry properly, whether it be numerically exact or approximate. The theorems are as follows. I . The transition vector cannot belong to a degenerate representation of the point group of the transition state. 2. transition vector must be svmmetric with re-. The ~~. spect to a symmetry operation of the transition state which leaves reactants or products unchanged. 3. The transition vector must be antisymmetric with respect to a symmetry operation of the transition state which converts reactants into products. The first theorem follows immediately from the requirement that at the transition state F must have one and only one negative eigenvalue.3 The proofs of the ~
~~
~
~~~
( I ) 1. N. Murrell and K. I. Laidler, Trans. Fareday Soc., 64, 311 (1968); see also 1. N. Murrell and G. L. Pratt, ibid.. 66, 1680 (1970). We have assumed that the potential surface is quadratic in the vicinity of the transition state. i.e.. that the transition vectoc doer not correspond to a zero eigenvalue of the force constant matrix F. Aside front pure translations and rotations, it is unlikely that the first and second derivatives of the potential energy will simultaneously vanish along an eigenvector of F. This argument is essentially the same as that which underl .i.p ~the . .... nnnc~ossinerule. K. R. Naavi and W. B. Brown, I n ( . J . ~~~~
~~
Quonfum Chem., 6,271 (1972). (2) E. B. Wilson, Jr., 1. C. Decius, nnd P. C. Cross, "Molecular Vibrations." McCraw-Hill. New York. N. Y., 1955. pp 106107.
November 29, I972
8619 R
t
P
T U‘
P‘
Figure 1. Illustration of proof of theorem 3. x4
second and third theorems are similar to one another and proceed by reductio ad absurdum. Here we outline the proof of theorem 3. The solid line in Figure 1 represents a path leading from reactants R through the transition state T to the products P. The path is chosen so that the energy at each point leading up to T is less than that at T itself. For simplicity we also require that the path follow the transition vector for a small but finite distance on each side of T. Let 6 be some symmetry operator4 of the transition state which converts reactants R into products P’ that differ from P by no more than free translations and rotations of isolated molecules. The application of 0 to the original path will convert each point into an energetically equivalent point on a path (dashed line in Figure 1) joining P’ and T. (The point T representing the transition state is, by definition, invariant under 0.) Let us suppose that in contradiction to theorem 3, the transition vector was symmetric under 0. It is evident from Figure 1 that if this were the case, then one could proceed from R to P’ to P by crossing from the solid path to the dashed path where they meet at the arrow. At no point would the energy be as high as that of the transition state. Since this contradicts the supposition that T is the lowest barrier, the transition vector cannot be symmetric. Since by theorem 1 it cannot belong to a degenerate representation, it can only be antisymmetric. This completes the proof. There are many reactions which possess some symmetry and thus to which one or more of the theorems can be applied. Here we restrict ourselves to a few rather striking examples. Consider the simple isoX3X4 -+ X1X4 topic exchange reaction XIXz X2X3(e.g., H2 Dz + 2HD). Among the symmetric structures proposed for the transition state are the tetrahedral (point group Td), centered equilateral tri-
+
+
+
(3) If the force constant matrix were to have a degenerate negative eigenvalue, then the transition state would resemble a hilltop, clearly not the lowest barrier between any two points. Transition states possessing a Jahn-Teller instability due to electronic degeneracy can be eliminated on similar grounds. The surface in this case differs from that just discussed only in that the top of the hill is sharpened into a peak rather than being rounded. See G. Herzberg, “Electronic Spectra of Polyatomic Molecules,” Van Nostrand-Reinhold, New York, N. Y., 1966, p 40 f f ; and J. N. Murrell, J . Chem. SOC.,Chem. Commun., 1044 (1972). (4) The mathematical effect of a symmetry operation on a molecular configuration is to “rotate” the entire molecule in space and then permute the nuclei back to the vicinity of their original positions. When applied to a symmetrical configuration, it leaves the structure unchanged; when applied to a distorted configuration, it changes it into a diferent, yet equivalent, distorted configuration. Pictorially this is equivalent to a rotation of the displacement vectors leading from the symmetric to the distorted configuration without changing the atomic labels (see Figure 2 for an example). This point is not always made clear in elementary treatments. For further discussion, see R. McWeeny, “Symmetry, An Introduction to Group Theory and Its Applications,” Macmillan, New York, N. Y . , 1963, pp 144-145.
x 1x 2 +
x 1 x 4 + x2x3
X3X4
(a)
(bl
Figure 2. Conversion of reactants into products by the tetrahedral operation S4.
angle (&) and square-planar ( D 4 h ) configuration^.^ The Td structure is illustrated in Figure 2. The arrows in the figure do not represent the transition vector but rather the displacements of the atoms to reactants or products. These arrows are the objects of the symmetry operations. The atomic labels remain un~ h a n g e d . ~In the case shown, the improper rotation g4 converts the reactant arrows (a) into a product set (b). Theorem 3 requires, therefore, that the transition vector be antisymmetric under g4, Inspection of the character table for the point group Td reveals that only the species Af and T2 behave in this way. The T2 representation is eliminated as the species of the transition vector by theorem 1, and standard group theoretical techniques can be used to show that A, is not available t o the internal coordinates of this system. We conclude, then, that the tetrahedral transition state is forbidden. Even if the AS species were available, the transition state would still be forbidden. This is because there are, in addition to the S4 operation, four distinct C3 operations which convert reactants into products, and the character table shows that A2 is symmetric under these operations. There are, incidentally, four more C3 operations which convert X2X4. reactants into the alternative products X1X3 It follows that the tetrahedral transition state is forbidden for this resction as well. By similar arguments, we can eliminate the D3h transition state for these reactions. The square-planar structure, on the other hand, is “allowed”; if it is the transition state, the theorems imply that the transition vector belongs to the B1, representation. In the case of the H2 D2 + 2HD exchange, it is likely that the Td and Dsh transition states could have been eliminated on the basis of the Jahn-Teller theorem since the relevant electronic states are probably degenerate.3J In other cases it may be difficult to decide whether or not such a degeneracy exists without computation. The theorems presented here, however, will generally make such a computation unnecessary. A
,.
+
+
( 5 ) E.y.,C . W. Wilson, Jr., and W. A. Goddard, J . Chem. Phys., 51, 716 (1969).
Commiinications to the Editor
-
8620
structure which is forbidden by our theorems will not be made allowed by additional instabilities. As a practical matter, we wish to point out that in all point groups containing a three-, five-, or sevenfold axis, the nondegenerate representations are symmetric with respect to rotations about this axis. Thus, from theorems 1 and 3, no structure for which rotation about this axis converts reactants into products can be a transition state. The Td and D3h structures discussed above fall into this category. Another example would be the trigonal-bipyramid structure for the substitution (1). Although this structure cannot be the transition
sition state for a b, then there must exist a lower energy transition state ( o f different symmetry) for the reaction a c. From this last example it should be clear that any of the “forbidden” structures discussed above could be stationary points with a single, nondegenerate, negative eigenvalue of F. Theorem 3 eliminated them as transition states only for the reactants and products under consideration. Acknowledgments. We thank Professor C. D. Ritchie for many useful discussions and the donors of the Petroleum Research Fund, administered by the American Chemical Society, for financial support. James W. McIver, Jr.* Department of Chemistry, State University of New York at Buffalo Buffalo, New York 14214
., I
+
X,--M-XA
I
Xj
Richard E. Stanton Department of Chemistry, Canisius College Buffalo, New York 14208 Received June I , 1972
(1)
x4
state, it could be a metastable intermediate (for which F would have no negative eigenvalues). There is some evidence to indicate-that this is in fact the case in at least one examplea6 The case of an evenfold axis is more complex. In reaction 2, for example, there are three distinct prod-
Thermal Cis-Trans Isomerization of Butadiene Sir : Many cis-trans isomerization reactions are now believed to proceed via a twisting mechanism which includes as a midpoint a species which has been termed a 1,2 diradical.’ We wish to report that a portion of the
H
H
D
D
w /7
X6
U
D
H
thermal cis-trans isomerization reactions of the 1,4dideuterio-1 ,3-butadienes2 also proceeds by such a mechanism, paths A and B, below, but that a correlated double isomerization, path C, appears to dominate the reaction mechanism6
X, b
H + D DH c
1
3
d
uct sets b, c, and d which can arise from reactants a cia the D 4 h (or C,,) structure as the transition state. The transition state is “allowed” for the reactions a + b and a +-d. This is because there exists anondegenerate representation (&) which is antisymmetric under the operations which convert a into b or d and symmetric under the operations which leave a unchanged. On the other hand, it is “forbidden” for the reaction a 4 c since every nondegenerate representation is symmetric under the operation Cd2which concerts a into c. Moreover, from the proof of theorem 3, it follows that if the D i hstructure is indeed the tran(6) F. Basolo and R. G. Pearson, “Mechanisms of Inorganic Reactions,” Wiley, New York, N. Y . , 1967, p 379.
Journal of the American Chemical Society
94:24
(1) S. W. Benson, “Thermochemical Kinetics,” Wiley, New York, N. Y . ,1968, p 72. (2) The trans,trans- and cis&-dideuteriobutadienes were prepared in good yield from the trans,trans-3 and cis,cis-dichlorobutadienes4 (pure by vpc) aia Zn-Cu couple reduction in refluxing dioxane-D20. The details of these procedures will be reported elsewhere. The cis,trans-dideuteriobutadienecould also be prepared from the corresponding dichlorobutadiene.‘ or more conveniently by the method of Fleming.& The structures and deuterium contents were confirmed by nmr and mass spectral measurements. The analysis of the pyrolysis products was most conveniently accomplished by a least-squares fit to five Raman lines, at 1171, 1216, 1226, 2260, and 2280 cm-’. (3) R. Huisgen, et a!., Angew. Chem., Int. E d . Engl., 7 5 , 585 (1966). Pure trans,trans-dichlorobutadienewas obtained after t w o recrystallizations from pentane. (4) Prepared and purified according to the method of P. D. Bartlett and G. E. H. Wallbillich, J . Amer. Chem. SOC.,91,409 (1969). ( 5 ) I. Fleming and E. Wildsmith, J . Chem. Soc. D,223 (1970). (6) A discussion of this possibility has appeared: J. I . Brauman a n d W. C. Archie, Jr., J . Amer. Chem. SOC.,94,4262 (1972).
1 Nocember 29, I972
